Aut(T_{n,r}) is exactly the bi-synchronizing transducer subgroup T B_{n,r} that preserves the cyclic order, and Out(T_{n,r}) contains a copy of Thompson's group F for n ≥ 3.
The growth rates of automaton groups generated by reset automata
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abstract
We give sufficient conditions for when groups generated by automata in a class $\mathcal{C}$ of transducers, which contains the class of reset automata transducers, have infinite order. As a consequence we also demonstrate that if a group generated by an automata in $\mathcal{C}$ is infinite, then it contains a free semigroup of rank at least 2. This gives a new proof, in the context of groups generated by automaton in $\mathcal{C}$, of a result of Chou showing that finitely generated elementary amenable groups either have polynomial growth or contain a free semigroup of rank at least 2.
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Automorphisms of the generalised Thompson's group $T_{n,r}$
Aut(T_{n,r}) is exactly the bi-synchronizing transducer subgroup T B_{n,r} that preserves the cyclic order, and Out(T_{n,r}) contains a copy of Thompson's group F for n ≥ 3.